Introduction to Polynomials - Algebra I

Introduction to Polynomials

Description: This lesson introduces polynomials, covering their definitions, types, and basic operations. Understand how to identify and work with polynomials through examples and exercises.

Outline

Definition of Polynomials
Types of Polynomials
Polynomial Operations
Examples and Exercises
Homework and Revision

Definition of Polynomials

A polynomial is an algebraic expression consisting of variables and coefficients, using only the operations of addition, subtraction, and multiplication, and non-negative integer exponents of variables. For example, 3x^2 + 2x - 5 is a polynomial.

Types of Polynomials

Monomial: A polynomial with one term. Example: 4x^3
Binomial: A polynomial with two terms. Example: 3x^2 - 2x
Trinomial: A polynomial with three terms. Example: x^2 + 5x + 6
Polynomial with more than three terms: Example: 2x^3 + 4x^2 - x + 7

Polynomial Operations

Polynomials can be added, subtracted, multiplied, and divided. Here are some key operations:

Addition: Combine like terms.
Subtraction: Subtract coefficients of like terms.
Multiplication: Use the distributive property.
Division: Divide terms of the polynomials.

Examples

Example 1

Problem: Add the polynomials 3x^2 + 4x - 5 and 2x^2 - 3x + 6.

Solution: Combine like terms to get 5x^2 + x + 1.

Example 2

Problem: Subtract 4x^3 - 2x + 7 from 6x^3 + x - 3.

Solution: The result is 2x^3 + 3x - 10.

Example 3

Problem: Multiply (x + 2) and (x - 3).

Solution: The product is x^2 - x - 6.

Example 4

Problem: Divide 4x^2 - 8 by 2x.

Solution: The quotient is 2x - 4/x.

Example 5

Problem: Simplify 5x^3 + 3x^2 - 2x + 4x^3 - 6x^2 + 2.

Solution: Combine like terms to get 9x^3 - 3x^2 - 2x + 2.

Example 6

Problem: Evaluate the polynomial 2x^2 - 3x + 4 for x = 2.

Solution: Substitute x = 2 to get 2(2)^2 - 3(2) + 4 = 8 - 6 + 4 = 6.

Exercises

Exercise 1

Problem: Add the polynomials 7x^2 - 3x + 4 and -2x^2 + 5x - 1.

Solution: Combine like terms to get 5x^2 + 2x + 3.

Exercise 2

Problem: Subtract 6x^2 + 4x - 5 from 8x^2 - 3x + 2.

Solution: The result is 2x^2 - 7x + 7.

Exercise 3

Problem: Multiply (2x - 3) and (x + 4).

Solution: The product is 2x^2 + 5x - 12.

Exercise 4

Problem: Divide 6x^3 - 4x by 2x.

Solution: The quotient is 3x^2 - 2.

Exercise 5

Problem: Simplify 4x^3 - x^2 + 2x + 7 - (x^3 - 3x^2 + x - 5).

Solution: Combine like terms to get 3x^3 + 2x^2 + x + 12.

Exercise 6

Problem: Evaluate the polynomial -3x^2 + 2x - 1 for x = -1.

Solution: Substitute x = -1 to get -3(-1)^2 + 2(-1) - 1 = -3 - 2 - 1 = -6.

Homework

Complete the following problems:

Problem 1: Add the polynomials 5x^2 - 2x + 3 and -x^2 + 4x - 6.
Problem 2: Subtract 7x^2 + 2x - 1 from 9x^2 - 3x + 4.
Problem 3: Multiply (x - 2) and (x + 5).
Problem 4: Divide 8x^3 - 2x by 4x.

Revision

Review the following points:

Understand the definition and types of polynomials.
Practice polynomial operations: addition, subtraction, multiplication, and division.
Simplify polynomials by combining like terms.